If the distance of a point from a fixed line is fixed , why do we say that the locus of the point is a line parallel to the given line ? The point also lies on a line which is perpendicular to the given line right ? Please ask me to elaborate with an example if the statement is not clear. Thank you.
Ok, so excuse the bad drawing. But imagine we tie a rope to a broomstick of length $d$ (for distance). Then the end of this rope (assuming it remain taut) has a few degrees of freedom. It can move along the length of the broomstick (the black line), it can also move around the broomstick. This process, if you were to consider the collection of all places where the green point can go, creates a cylinder around the broom.
If we were to cut right through the middle of this broom we lose one degree of freedom: going "around" the broom. So we're left with two lines whose distance - at every point - from the broom is $d$. This is the definition of a parallel line.
The key understanding here, is that the "locus" is the collection of all points for which a certain condition holds (a distance $d$ from the line)